Skip to main content
SpringerLink
Log in

Hyperstability results for a generalized radical cubic functional equation related to additive mapping in non-Archimedean Banach spaces

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

The aim of this paper is to introduce and solve the following generalized radical cubic functional equation

$$\begin{aligned} f\left( \root 3 \of {\sum _{i=1}^{k}x_{i}^{3}}\right) =\sum _{i=1}^{k}f(x_{i}),\quad k\in \mathbb {N}_{2}. \end{aligned}$$

We also investigate some hyperstability results for this equation in non-Archimedean Banach spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aiemsomboon, L., Sintunavarat, W.: On generalized hyperstability of a general linear equation. Acta Math. Hungar. 149, 413–422 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aiemsomboon, L., Sintunavarat, W.: On a new type of stability of a radical quadratic functional equation using Brzdȩk’s fixed point theorem. Acta Math. Hungar. 151, 35–46 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alizadeh, Z., Ghazanfari, A.G.: On the stability of a radical cubic functional equation in quasi-\(\beta \)-spaces. J. Fixed Point Theory Appl. 18, 843–853 (2016)

    Article  MathSciNet  Google Scholar 

  4. Almahalebi, M., Chahbi, A.: Hyperstability of the Jensen functional equation in ultrametric spaces. Aequat. Math. 91, 647–661 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bahyrycz, A., Piszczek, M.: Hyperstability of the Jensen functional equation. Acta Math. Hungar. 142, 353–365 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bourgin, D.G.: Classes of transformations and bordering transformations. Bull. Am. Math. Soc. 57, 223–237 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brzdȩk, J.: Remarks on hyperstability of the Cauchy functional equation. Aequat. Math. 86, 255–267 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brzdȩk, J.: Hyperstability of the Cauchy equation on restricted domains. Acta Math. Hungar. 141, 58–67 (2013)

    Article  MathSciNet  Google Scholar 

  10. Brzdȩk, J.: A hyperstability result for the Cauchy equation. Bull. Aust. Math. Soc. 89, 33–40 (2014)

    Article  MathSciNet  Google Scholar 

  11. Brzdȩk, J.: Remarks on solutions to the functional equations of the radical type. Adv. Theory Nonlinear Anal. Appl. 1, 125–135 (2017)

    Google Scholar 

  12. Brzdȩk, J.: Remark 3. In: Report of Meeting of 16th International Conference on Functional Equations and Inequalities (Bȩdlewo, Poland, May 17–23, 2015), p. 196, Ann. Univ. Paedagog. Crac. Stud. Math. 14, 163–202 (2015)

  13. Brzdȩk, J., Ciepliñski, K.: A fixed point approach to the stability of functional equations in non-Archimedean metric spaces. Nonlinear Anal. 74, 6861–6867 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Brzdȩk, J., Ciepliński, K.: Hyperstability and superstability. Abstr. Appl. Anal. 2013, Art. ID 401756 (2013)

  15. Gordji, M.E., Khodaei, H., Ebadian, A., Kim, G.: Nearly radical quadratic functional equations in \(p\)-2-normed spaces. Abstr. Appl. Anal. 2012, Art. ID 896032 (2012)

  16. Gordji, M.E., Parviz, M.: On the Hyers–Ulam stability of the functional equation \(f\left(\root 2 \of {x^{2}+y^{2}}\right)=f(x)+f(y)\). Nonlinear Funct. Anal. Appl. 14, 413–420 (2009)

    MathSciNet  Google Scholar 

  17. Gajda, Z.: On stability of additive mappings. Int. J. Math. Math. Sci. 14, 431–434 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Găvruţa, P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gselmann, E.: Hyperstability of a functional equation. Acta Math. Hungar. 124, 179–188 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  21. Khodaei, H., Gordji, M.E., Kim, S., Cho, Y.: Approximation of radical functional equations related to quadratic and quartic mappings. J. Math. Anal. Appl. 395, 284–297 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Khrennikov, A.: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer Academic Publishers, Dordrecht (1997)

    Book  MATH  Google Scholar 

  23. Kim, S., Cho, Y., Gordji, M.E.: On the generalized Hyers–Ulam–Rassias stability problem of radical functional equations. J. Inequal. Appl. 2012, 186 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Maksa, G., Páles, Z.: Hyperstability of a class of linear functional equations. Acta Math. Acad. Paedag. Nyíregyháziensis 17, 107–112 (2001)

    MathSciNet  MATH  Google Scholar 

  25. Piszczek, M.: Remark on hyperstability of the general linear equation. Aequat. Math. 88, 163–168 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rassias, T.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rassias, T.M., Šemrl, P.: On the behavior of mappings which do not satisfy Hyers–Ulam stability. Proc. Am. Math. Soc. 114, 989–993 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ulam, S.M.: Problems in Modern Mathematics. Wiley, New York (1964). Science Editions

    MATH  Google Scholar 

Download references

Acknowledgements

C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2017R1D1A1B04032937).

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, Ibn Tofail University, BP 14000 , Kenitra, Morocco

    Muaadh Almahalebi, Ahmed Charifi & Samir Kabbaj

  2. Research Institute for Natural Sciences, Hanyang University, Seoul, 04763, Republic of Korea

    Choonkil Park

Authors
  1. Muaadh Almahalebi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ahmed Charifi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Choonkil Park
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Samir Kabbaj
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Choonkil Park.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Almahalebi, M., Charifi, A., Park, C. et al. Hyperstability results for a generalized radical cubic functional equation related to additive mapping in non-Archimedean Banach spaces. J. Fixed Point Theory Appl. 20, 40 (2018). https://doi.org/10.1007/s11784-018-0524-7

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s11784-018-0524-7

Keywords

Mathematics Subject Classification

Search

Navigation